1-1-1 1-1-3 1-4 1-5 1-6 1-7 1-8
(differential equation) F(,, ') = (first order differential equation) ' + + 4= 1 + = e
hapter 1 (separable equation) A( ) d = B( ) d (1.1) ò ò Ad ( ) = Bd ( ) 1 - cos = d = cos d = sin + = sin = sin + 1 = sin + 3 1.1 3
(general solution) (initial condition) (boundar condition) () = = = sin + (particular solution) ( ) = 1 sin + = 1 = 1 = sin + 1 3 = 4 + 1 () = 3 3 d = (4 + 1) d ò 3 = (4 + 1) d = + + 4 () = 3 3= + + \ = 3 4 = + + 3 4
hapter 1 3 - = (1) = d = d d = d 1 = + = 1+ = 1 1 1 = + 4 d 3 ( ) d = + () = 8 5
d = 3 + d ln 3 + = + ln e e ln + = + 3 3 3 + * e = e e = e 3 = * e - * 8= * - \ * = 1 3 = 1e - 5 (1,1) - 6
hapter 1 - = (1) = 1 ln =- ln + = e = e - + ln ln = ln e e - * = / = * = 1 = 1 1 = = 1 6 P V(P) V - (Bole-Maritte's law for ideal P gas) 7
dv V =- dp P dv V dp =- P ln V =- ln P + V = e = e = e e * = P PV = * ln V - ln P + -ln P 7 - = (1.1) 8
hapter 1 1. - = Ans ln = +. - = 1 Ans = + 3. + = 1 1 Ans + = 4. - = Ans ln - = 5. - = Ans = + 6. d 3 + ( + 1) d= Ans 3 4 + 4( + 1) 3 = 7. ( + 1) d + ( - 1) d = 1 1 Ans + + ln - 1 + - + ln + 1 = 8. 4 - = Ans 4 = 9. (1) () = 3 ( + ) (1) = 8 Ans 1 3 1 + = e - + = (1) = Ans 1 1 1 = 3 + - 3 3 9
d 1 (3) = (4) = 8 d 1 1 19 Ans + = 4 1. ( + 1) + + 1= () = 1 Ans 1 - = tan a+ tanb tan( a+ b) = 1 + 1 - tana + tanb 3 11. (1 + ) d- d= (1) = Ans 3 = 4(1 + 3 ) -3 1. = + 4 Ans ( + 4) =- 3 + + 3 + 8= 8 + 1 13. = (1) = 5 Ans 4 65 = 4 + + 4 4 14. - e = (1) = 7 - -7 Ans - e = ln - e d 15. e d = + () = 3 Ans 3 - - =-e - 3e + 1 3 1
hapter 1 (equation reducible to separable form) æ ö = f ç çè ø (1.) (1.) u = (1.3) ( ) u ( ) = u = u+ u (1.4) (1.) u+ u = f( u) u = f( u) - u (1.1) du d = f ( u) -u (1.5) u = 11
8 = 1+ (1.) u = = u + u u+ u = 1+ u u = 1 d du = u= ln + u = ln = + = ln + 9-3 = (1.3)(1.4) u + u- 3u= 1
hapter 1 du = ( u + 1) d du d = u+ 1 ln u+ 1 = ln + * u+ 1= (1.3) 1 + = 1 - + = æ ö - ç + = çè ø (1.3)(1.4) uu ( + u) - u + = 13
u + uu - u + = u + =-uu du =-u d udu d =- + u ln ln + u =- + * + u = u æ ö + ç = çè ø + = 14
hapter 1 11 = + æ ç 1 ö = çè 4 ø = + (1.3)(1.4) u+ u = + u du d = + u ln + u = ln + * + u= u + = = - ç æ 1 ö = çè 4 ø = 8 - = 8 15
1 æ ö = 1- + ç çè ø (1.3) (1.4) u+ u = 1- u+ u u = 1- u+ u = ( u- 1) du ( u-1) d = - 1 = ln + u -1 ( u- 1)(ln + ) =- 1 u = æ ö ç - 1 (ln + ) =-1 çè ø (1.) 13 14 16
hapter 1 13 (- 4+ 5) + - + 4 = - = u -u = 1 = (1 - u ) é 1 ù (u+ 5) (1- u ) + u+ 4 = êë úû (u+ 5)(1 - u ) =- ( u+ 4) (u+ 5) u = 4u+ 13 æ 3 ö ç 1- du = d çè 4u + 13 ø 3 u- ln 4 u+ 13 = + 4 u= - (-4) 4 + 8+ 3ln 4-8+ 13 = 17
14 - e + = = u u u u = u = - = u u u e - + = -u u edu = d u e = + e = + = ln + 1 = ln + 18
hapter 1 1. - = Ans ln = +. - + = Ans ln = + 3. + =- 4 æ ö Ans ln ç + =- ln + çè ø 4. + = 3 æ ö Ans ln ç - 1 =- 3ln + çè ø 5. - = 1 Ans ln = + 6. ( + ) d+ (+ 3 ) d= Ans + 4 + 3 = 7. ( 3 + 3 ) d + 3 d = é æ ö Ans = 1 4 + ç çè ê ø ë 1 3-4 ù úû 19
8. ( - ) d + d = Ans - 4 = 9. ( + ) d + (3 + 3-4) d = + = t Ans + 3+ ln -- = 1. = ( - ) - = u 1+ ce Ans = + 1 - ce 11. + = e - + = u Ans + e = 1. ( + ) = Ans ln - = 13. + 3= æ Ans 1 4 ö = ç - çè ø 1-4 æ ö 14. - = sec ç çè ø -1 Ans = sin ( + )